Properties

Label 1-1048-1048.523-r0-0-0
Degree $1$
Conductor $1048$
Sign $1$
Analytic cond. $4.86689$
Root an. cond. $4.86689$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 7-s + 9-s + 11-s − 13-s − 15-s − 17-s − 19-s − 21-s + 23-s + 25-s + 27-s + 29-s + 31-s + 33-s + 35-s + 37-s − 39-s + 41-s + 43-s − 45-s + 47-s + 49-s − 51-s − 53-s − 55-s + ⋯
L(s)  = 1  + 3-s − 5-s − 7-s + 9-s + 11-s − 13-s − 15-s − 17-s − 19-s − 21-s + 23-s + 25-s + 27-s + 29-s + 31-s + 33-s + 35-s + 37-s − 39-s + 41-s + 43-s − 45-s + 47-s + 49-s − 51-s − 53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1048\)    =    \(2^{3} \cdot 131\)
Sign: $1$
Analytic conductor: \(4.86689\)
Root analytic conductor: \(4.86689\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1048} (523, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1048,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.584221912\)
\(L(\frac12)\) \(\approx\) \(1.584221912\)
\(L(1)\) \(\approx\) \(1.183859034\)
\(L(1)\) \(\approx\) \(1.183859034\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
131 \( 1 \)
good3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.56718153107034037495030481434, −20.52368137868376890796363248692, −19.665799159285760822903372773771, −19.425821237939540728397202277026, −18.923410400316654056026151842910, −17.60127961710355694346731329502, −16.721479416003325235951365179, −15.86276451950599134392655731391, −15.21209697194462697533507014942, −14.61204931535224935017667927903, −13.69044827075253557429412273804, −12.71184959189750184204693100485, −12.30016859659436169856441318583, −11.16727150803664807865363283781, −10.21306790244152565779316634171, −9.23351424705048732396454081212, −8.79493319372769948409627095293, −7.75860514252762772599658295166, −6.95302990726421847670956160160, −6.35530651971128876955491550170, −4.52351020293829141410291409938, −4.17123136250834920402394989016, −3.05062975092852984149397841624, −2.42575377988817026551361968070, −0.85251316093032440677723859792, 0.85251316093032440677723859792, 2.42575377988817026551361968070, 3.05062975092852984149397841624, 4.17123136250834920402394989016, 4.52351020293829141410291409938, 6.35530651971128876955491550170, 6.95302990726421847670956160160, 7.75860514252762772599658295166, 8.79493319372769948409627095293, 9.23351424705048732396454081212, 10.21306790244152565779316634171, 11.16727150803664807865363283781, 12.30016859659436169856441318583, 12.71184959189750184204693100485, 13.69044827075253557429412273804, 14.61204931535224935017667927903, 15.21209697194462697533507014942, 15.86276451950599134392655731391, 16.721479416003325235951365179, 17.60127961710355694346731329502, 18.923410400316654056026151842910, 19.425821237939540728397202277026, 19.665799159285760822903372773771, 20.52368137868376890796363248692, 21.56718153107034037495030481434

Graph of the $Z$-function along the critical line